Optical scanning system

Abstract

An optical scanning system includes a variable-focus element, an imaging lens and a deflector, wherein the reciprocal of the focal length f of the variable-focus element is changed from 1/f.sub.MIN to 1/f.sub.MAX, and for the case that the equation
1/f={(1/f.sub.MAX)+(1/f.sub.MIN)}/2
holds, a beam which has passed through the variable-focus element is a divergent beam, and $\begin{array}{cc}{x}_2+\frac{x_2^2}{x_1}>x_3& \left(1\right)\end{array}$
is satisfied, where x.sub.1 represents a distance from a virtual image point of the divergent beam to the principal point on the entry side of the variable-focus element, x.sub.2 represents a distance from the principal point on the exit side of the variable-focus element to the principal point on the entry side of the imaging lens, and x3 represents a distance from the principal point on the exit side of the imaging lens to an image point.

Claims

1. An optical scanning system for scanning a plane with a beam emitted by a light source, the system comprising a variable-focus element, an imaging lens and a deflector, wherein the reciprocal of the focal length f of the variable-focus element is changed from the minimum value 1/f.sub.MIN to the maximum value 1/f.sub.MAX, and for the case that the equation
1/f={(1/f.sub.MAX)+(1/f.sub.MIN)}/2 holds, the beam which has passed through the variable-focus element is a divergent beam at a point just behind the variable-focus element, and $\begin{array}{cc}{x}_2+\frac{x_2^2}{x_1}>x_3& \left(1\right)\end{array}$ is satisfied, where x.sub.1 represents a distance from a virtual image point of the divergent beam to the principal point on the entry side of the variable-focus element, x.sub.2 represents a distance from the principal point on the exit side of the variable-focus element to the principal point on the entry side of the imaging lens and x.sub.3 represents a distance from the principal point on the exit side of the imaging lens to an image point.

2. An optical scanning system according to claim 1, wherein
|x.sub.1|<3.2x.sub.3(2)
and
x.sub.2<0.8x.sub.3(3) are further satisfied.

3. An optical scanning system according to claim 1, wherein when f is defined as $\begin{array}{cc}f=\frac{2}{\frac{1}{f_{\max }}-\frac{1}{f_{\min }}},& \left(4\right)\\ 0.035<\frac{x_1\left(x_1{x}_2+x_2^2-x_1{x}_3\right)}{{f\left(x_1+x_2\right)}^2}<0.5& \left(5\right)\end{array}$ is further satisfied.

4. An optical scanning system according to claim 1, wherein the plane is a flat plane and when a height from the center to each end along a scanning line on the flat plane is represented as H, a focal length of the imaging lens in a cross section of the main scanning direction, in which the beam moves for scanning, is represented as f.sub.2, and f is defined as $\begin{array}{cc}f=\frac{2}{\frac{1}{f_{\max }}-\frac{1}{f_{\min }}},& \left(4\right)\\ .Math.-\frac{{\left(x_{3-}-x_{3+}\right)}^2}{H^2}+\frac{x_1\left(f_2-x_2\right)}{f\left(x_1+x_2-f_2\right)}.Math.<0.05& \left(6\right)\end{array}$ is further satisfied, where $\begin{array}{cc}{x}_{3+}=\frac{f_2^2\left(x_1-f\right)}{\left(x_1+x_2-f_2\right)\left(x_1-f\right)-x_1^2}+f_2\mathrm{and}& \left(7\right)\\ x_{3-}=\frac{f_2^2\left(x_1+f\right)}{\left(x_1+x_2-f_2\right)\left(x_1+f\right)-x_1^2}+f_2.& \left(8\right)\end{array}$

5. An optical scanning system according to claim 1, further comprising a condenser lens that is installed on the light source side of the imaging lens and is configured to move in the direction of the optical axis such that the beam forms an image on the plane, wherein provided that the condenser lens and a virtual variable-focus element are fixed at the center of the moving range, the minimum value 1/f.sub.MIN and the maximum value 1/f.sub.MAX of the reciprocal of the focal length f of the virtual variable-focus element are determined such that a change of the position of the image caused by the movement of the condenser lens is reproduced and the virtual variable-focus element is regarded as the variable-focus element.

6. An optical scanning system according to claim 1, wherein the focal length of the imaging lens in the main scanning direction and the focal length of the imaging lens in the sub scanning direction are different from each other.

Description

BRIEF DESCRIPTION OF DRAWINGS

(1) FIG. 1 shows an arrangement of the optical scanning system 100A of Example 1.;

(2) FIG. 2A shows the state in which f=f.sub.MEAN;

(3) FIG. 2B shows the state in which f=f.sub.MAX;

(4) FIG. 2C shows the state in which f=f.sub.MIN;

(5) FIG. 3 illustrates how the variable-focus element performs positional corrections of the image surface;

(6) FIG. 4 shows an arrangement of an optical scanning system of Example 2;

(7) FIG. 5 shows an arrangement of an optical scanning system of Example 3;

(8) FIG. 6 shows an arrangement of an optical scanning system of Comparative Example; and

(9) FIG. 7 shows a relationship between position and diameter of the spot formed on the light receiving surface by the beam.

DESCRIPTION OF EMBODIMENTS

(10) FIG. 1 shows an arrangement of an optical scanning system 100A according to an embodiment of the present invention. The embodiment corresponds to Example 1 that will be described later. The optical scanning system 100A includes a condenser lens 103A, an aperture 104A, a variable-focus element 105A, an imaging lens 106A and a deflector 107 such as a polygon mirror. Light emitted by a laser diode light source 101, wavelength of which is 780 nm, is converted into a predetermined divergent beam by the condenser lens 103A, made to pass through the variable-focus element 105A and the imaging lens 106A and deflected by the deflector 107 such that the beam scans a light receiving surface 108 in the horizontal direction in FIG. 1. The variable-focus element 105A is controlled such that the beam constantly forms an image on the receiving surface during the period of the scanning. In FIG. 1, the center of the scanning line on the light receiving surface 108 is represented as C and the both ends are represented as E.

(11) FIGS. 2A, 2B and 2C illustrate how the variable-focus element functions. In FIGS. 2A, 2B and 2C, the condenser lens, the aperture and the deflector have been omitted. In FIGS. 2A, 2B and 2C, P1 represents the virtual image point of the divergent beam, P2 represents the principal point on the entry side of the variable-focus element 105, P3 represents the principal point on the exit side of the variable-focus element 105, P4 represents the principal point on the entry side of the imaging lens 106, P5 represents the principal point on the exit side of the imaging lens 106, and P6 represents the image point of the beam.

(12) It is assumed that the reciprocal of the focal length f of the variable-focus element 105 is changed from the minimum value 1/f.sub.MIN to the maximum value 1/f.sub.MAX such that the beam constantly forms an image on the light receiving surface during the period of the scanning. When
1/f={(1/f.sub.MAX)+(1/f.sub.MIN)}/2=1/f.sub.MEAN,
the distance from the virtual image point P1 of the divergent beam to the principal point P2 on the entry side of the variable-focus element 105 is represented as x.sub.1, the distance from the principal point P3 on the exit side of the variable-focus element 105 to the principal point P4 on the entry side of the imaging lens 106 is represented as x.sub.2, and the distance from the principal point P5 on the exit side of the imaging lens 106 to the image point P6 is represented as x.sub.3.

(13) FIG. 2A shows the state in which f=f.sub.MEAN.

(14) FIG. 2B shows the state in which f=f.sub.MAX. In this state, the distance from the principal point P5 on the exit side of the imaging lens 106 to the image point P6 is represented as x.sub.3+. x.sub.3+ corresponds to the distance along the path of the principal ray of the beam from the principal point on the exit side of the imaging lens 106A to the center C of the scanning line on the light receiving surface 108.

(15) FIG. 2C shows the state in which f=f.sub.MIN. In this state, the distance from the principal point P5 on the exit side of the imaging lens 106 to the image point P6 is represented as x.sub.3. x.sub.3 corresponds to the distance along the path of the principal ray of the beam from the principal point on the exit side of the imaging lens 106A to each end E of the scanning line on the light receiving surface 108.

(16) Based on paraxial approximation of the simplified optical system shown in FIGS. 2A, 2B and 2C, how the optical scanning system according to the present invention functions will be described. It is assumed that the reciprocal of the focal length of the variable-focus element 105 ranges from 1/f (f corresponds to f=f.sub.MIN) to +1/f (+f corresponds to f=f.sub.MAX) when power of 0 (which corresponds to an afocal system) is used as a reference.

(17) NA on the virtual image side is represented as NAo, NA on the image side is represented as NAi, the magnification at the variable-focus element is represented as .sub.1, and the magnification at the imaging lens is represented as .sub.2. A relationship between NAi and NAo can be expressed by the following equations.

(18) $\begin{array}{cc}{\mathrm{NA}}_o=-{}_1{}_2{\mathrm{NA}}_i& \left(9\right)\\ \frac{{\mathrm{NA}}_i}{{\mathrm{NA}}_o}=\frac{1}{-{}_1{}_2}=M& \left(10\right)\end{array}$
M is a ratio of the numerical aperture NAi on the image side to the numerical aperture NAo on the virtual image side.

(19) When the focal length of the variable-focus element is represented as f and the focal length of the imaging lens is represented as f2, the magnification at the variable-focus element .sub.1 and the magnification at the imaging lens .sub.2 can be represented by the following equations. The second term of Equation (12) can be obtained by substitution of Equation (7) into the third term of Equation (12). Equation (7) will be described later.

(20) $\begin{array}{cc}{}_1=\frac{f}{x_1-f}& \left(11\right)\\ {}_2=\frac{{\mathrm{ff}}_2}{-{}_1{x}_1^2+f\left(x_1+x_2-f_2\right)}=\frac{x_3-f_2}{f_2}& \left(12\right)\end{array}$

(21) The ratio M obtained when the focal length of the variable-focus element is +f is represented as M.sub.+ and the ratio M obtained when the focal length of the variable-focus element is f is represented as M.sub.. A relative difference K between the ratio M.sub.+ at the center C of the scanning line and the ratio M.sub. at each end E of the scanning line can be represented by the following equation.

(22) 0$\begin{array}{cc}K=\frac{M_{-}-M_{+}}{M_{-}+M_{+}}& \left(13\right)\end{array}$
Using Equations (10) to (12), the above-described equation can be changed into the following one.

(23) $\begin{array}{cc}K=\frac{x_1\left(x_1{x}_2+x_2^2-x_1{x}_3\right)}{{f\left(x_1+x_2\right)}^2}& \left(14\right)\end{array}$

(24) K should be positive in order that NAi at each end E is greater than that at the center C. Since x.sub.1 and f is positive, the following expression has to be satisfied in order that K is positive.
x.sub.1x.sub.2+x.sub.2.sup.2x.sub.1x.sub.3>0(15)
The following expression can be obtained by changing Expression (15).

(25) $\begin{array}{cc}{x}_2+\frac{x_2^2}{x_1}>x_3& \left(1\right)\end{array}$

(26) For downsizing of the whole optical scanning system, reducing the distance x.sub.2 between the lenses is effective. However, when x.sub.1 is relatively great and the beam becomes close to a parallel beam immediately after passing through the variable-focus element, the value of x.sub.2 cannot be made smaller than that of x.sub.3. Accordingly, for downsizing of the apparatus, it is effective to reduce the value of x.sub.1 such that the beam is more widely diverged immediately after passing through the variable-focus element. By making the absolute value of x.sub.1 less than 3.2 times x.sub.3, x.sub.2 can be significantly reduced (0.8 times as great as x.sub.3).
|x.sub.1|<3.2x.sub.3(2)
x.sub.2<0.8x.sub.3(3)

(27) Because of the limited performance of the deflector and the like, the maximum incident angle of the scanning beam to the image plane is approximately 60 degrees at maximum. On the other hand, when the maximum incident angle is below 15 degrees, the optical path length and the size of the condenser lens become larger, and therefore the whole apparatus becomes larger. On this occasion, the range of the relative difference K concerning NAi, which is used for the correction can be expressed by the following Expression (16). In the range of K which does not satisfy Expression (16), an amount of correction is excessive and such an excessive amount of correction makes the spot diameter on the periphery excessively small.
0.035<K<0.5(16)

(28) Substituting Equation (14) in Expression (16) yields the following expression.

(29) $\begin{array}{cc}0.035<\frac{x_1\left(x_1{x}_2+x_2^2-x_1{x}_3\right)}{{f\left(x_1+x_2\right)}^2}<0.5& \left(5\right)\end{array}$

(30) The distance x.sub.3 from the principal point on the exit side of the imaging lens to the image point varies between x.sub.3+ and x.sub.3 as the focal length of the variable-focus element varies between +f and f.

(31) $\begin{array}{cc}{x}_{3+}=\frac{f_2^2\left(x_1-f\right)}{\left(x_1+x_2-f_2\right)\left(x_1-f\right)-x_1^2}+f_2& \left(7\right)\\ x_{3-}=\frac{f_2^2\left(x_1+f\right)}{\left(x_1+x_2-f_2\right)\left(x_1+f\right)-x_1^2}+f_2& \left(8\right)\end{array}$

(32) In general, when the focal length f of a lens and the distance x from an object to the principal point on the entry side of the lens are determined, the distance x from the principal point on the exit side of the lens to the image plane is detemined by the following equation.
f/(xf)=(xf)/f
Accordingly, in FIG. 2, a position of a first image formed by the variable-focus element, of the vertual image can be obtained, and then a position of a second image formed by the imaging lens 106, of the first image can be calculated. Thus, Equations (7) and (8) can be derived.

(33) FIG. 3 illustrates how the variable-focus element performs positional corrections of the image surface. In FIG. 3, l represents the distance from the point of reflection on the deflector to the center C of the scanning line along the path of the principal ray of the beam which reaches the center C of the scanning line. H represents the distance from the center C of the scanning line to each end E of the scanning line. Provided that the curverture of the image plane is corrected by the variable-focus element, the following equation holds.
(l+x.sub.3x.sub.3+).sup.2l.sup.2=H.sup.2(17)

(34) Accordingly, cos which represents a change of the spot diameter when the incidence angle of the beam onto the image plane (light receiving surface) is , can be expressed by the following equation by the use of Equation (17).

(35) $\begin{array}{cc}\cos =\frac{1}{1+x_{3-}-x_{3+}}=\frac{H^2-{\left(x_{3-}-x_{3+}\right)}^2}{H^2+{\left(x_{3-}-x_{3+}\right)}^2}& \left(18\right)\end{array}$

(36) A relative difference J of effect of oblique incidence for the beam which reaches a point on the scanning line between the center and the both ends can be expressed by the following equation by the use of Equation (18).

(37) $\begin{array}{cc}J=\frac{\cos -1}{\cos +1}=-\frac{{\left(x_{3-}-x_{3+}\right)}^2}{H^2}& \left(19\right)\end{array}$

(38) According to Expression B, the spot diameter is kept constant along the scanning line when an effect of increase of M, that is, an effect of increase of NA (a positive value of K) and an effect of decrease of cos caused by increase of the incidence angle balance each other, in other words, when the following equation holds.
J+K=0(20)

(39) When a spot diameter change up to 5% is allowable, the following expression should be satisfied.
|J+K|<0.05(21)
Substituting Equations (14) and (19) in Expression (21) yields the following expression.

(40) $\begin{array}{cc}.Math.-\frac{{\left(x_{3-}-x_{3+}\right)}^2}{H^2}+\frac{x_1\left(f_2-x_2\right)}{f\left(x_1+x_2-f_2\right)}.Math.<0.05& \left(6\right)\end{array}$

(41) Examples and a comparative example will be described below.

Example 1

(42) FIG. 1 shows an arrangement of the optical scanning system 100A of Example 1. The optical scanning system 100A includes the condenser lens 103A, the aperture 104A, the variable-focus element 105A, the imaging lens 106A and the deflector 107 such as a polygon mirror. Light emitted by the laser diode light source 101, wavelength of which is 780 nm, is converted into a predetermined divergent beam by the condenser lens 103A, made to pass through the variable-focus element 105A and the imaging lens 106A and deflected by the deflector 107 such that the beam scans a light receiving surface 108 in the horizontal direction in FIG. 1. The variable-focus element 105A is controlled such that the beam constantly forms an image on the light receiving surface during the period of the scanning. In FIG. 1, the center of the scanning line on the light receiving surface 108 is represented as C and the both ends are represented as E.

(43) Table 1 shows numerical data of the optical scanning system 100A of Example 1. As to space or thickness in Table 1 and the other tables, for example, space or thickness of the datum of light source represents a space between the datum of light source and the entry side surface of the condenser lens which is next to the light source and space or thickness of the entry side surface of the condenser lens represents a thickness of the condenser lens. Further, space or thickness is that along the optical path of the light which travels from the datum of light source and reaches the center C of the scanning line.

(44) The variable-focus element is a virtual one which has a thickness of 0. However, in FIG. 1, the variable-focus element 105A is drawn as one which has a predetermined thickness for the sake of better understanding.

(45) TABLE-US-00001 TABLE 1 Radius of Space or curvature thickness at center Item (mm) (mm) Material Datum of light source 4.148 Entry side surface of 2 17.751 BK7 condenser lens Exit side surface of 1 3.662 condenser lens Aperture plane 3 Principal point on entry 0 side of variable-focus element Principal point on exit 135 side of variable-focus element Entry side surface of 2 53.442 BK7 imaging lens Exit side surface of 23.235 Infinity imaging lens Deflecting mirror 206 Image surface of scanning beam (light receiving surface)

(46) The material of the condenser lens 103A and the imaging lens 106A is borosilicate glass, the brand name of which is BK7. For light of wavelength of 780 nm, the value of refractive index is 1.511 and the value of Abbe's number is 64.2.

(47) The entry side surface of the condenser lens 103A and the entry side and the exit side surfaces of the imaging lens 106A are spherical or flat. The exit side surface of the condenser lens 103A is defined by the following equations.

(48) $\begin{array}{cc}z=\frac{{\mathrm{cr}}^2}{1+\sqrt{1-}\left(1+k\right)c^2{r}^2}+\underset{n}{.Math.}{a}_{2n}{r}^{2n}& \left(22\right)\\ r=\sqrt{x^2+y^2}& \left(23\right)\end{array}$
r represents distance from the optical axis, and z represents distance in the direction of the optical axis from the intersection point between the optical axis and the exit side surface of the condenser lens 103A as the reference point. c represents curvature at the center and R represents the radius of curvature at the center. Further, a represents aspherical coefficients and n represents integers.

(49) In the examples and comparative example, the optical axis is the line connecting the center of each optical element and agrees with the path of the primary ray of the beam which reaches the canter C of the scanning line.

(50) Table 2 shows numerical data of the exit side surface of the condenser lens 103A of Example 1. The unit of length in the table is millimeter. The focal length of the condenser lens 103A is 6.1 millimeters.

(51) TABLE-US-00002 TABLE 2 Item Data (Coefficient) Radius of curvature 3.662 at center R = 1/c Conic constant k 0.997 Fourth-order aspherical 4.91E04 coefficient a4

Example 2

(52) FIG. 4 shows an arrangement of an optical scanning system 100B of Example 2. The optical scanning system 100B includes a condenser lens 103 B, an aperture 104B, an imaging lens 106B and a deflector 107 such as a polygon mirror. Light emitted by a laser diode light source 101, wavelength of which is 780 nm, is converted into a predetermined divergent beam by the condenser lens 103B, made to pass through the imaging lens 106B and deflected by the deflector 107 such that the beam scans a light receiving surface 108 in the horizontal direction in FIG. 4. The condenser lens 103B is made to move in the direction of the optical axis in a predetermined section in synchronization with the scanning caused by the deflector 107. Movement of the condenser lens 103B in the direction of the optical axis changes a position of an image of the beam along the optical path. The position of the condenser lens 103B is controlled such that the beam constantly forms an image on the light receiving surface during the period of the scanning. The distance that the condenser lens 103B moves in the direction of the optical axis is 229 micrometers. In FIG. 4, the center of the scanning line on the light receiving surface 108 is represented as C and the both ends are represented as E.

(53) Table 3 shows numerical data of the optical scanning system 100B of Example 2.

(54) TABLE-US-00003 TABLE 3 Radius of Space or curvature thickness at center Item (mm) (mm) Material Datum of light source 4.196 (4.105-4.334) Entry side surface of 2 17.751 BK7 condenser lens Exit side surface of 1 3.662 condenser lens Aperture plane 134 Entry side surface of 2 53.442 BK7 imaging lens Exit side surface of 23.235 Infinity imaging lens Deflecting mirror 206 Image surface of scanning beam (light receiving surface)

(55) The material of the condenser lens 103B and the imaging lens 106B is borosilicate glass, the brand name of which is BK7. For light of wavelength of 780 nm, the value of refractive index is 1.511 and the value of Abbe's number is 64.2.

(56) The entry side surface of the condenser lens 103B and the entry side and the exit side surfaces of the imaging lens 106B are spherical or flat. The exit side surface of the condenser lens 103B is defined by the following equations.

(57) $\begin{array}{cc}z=\frac{{\mathrm{cr}}^2}{1+\sqrt{1-}\left(1+k\right)c^2{r}^2}+\underset{n}{.Math.}{a}_{2n}{r}^{2n}& \left(22\right)\\ r=\sqrt{x^2+y^2}& \left(23\right)\end{array}$

(58) Table 4 shows numerical data of the exit side surface of the condenser lens 103B of Example 2. The unit of length in the table is millimeter. The focal length of the condenser lens 103B is 6.1 millimeters.

(59) TABLE-US-00004 TABLE 4 Item Data (Coefficient) Radius of curvature 3.662 at center R = 1/c Conic constant k 0.997 Fourth-order aspherical 4.91E04 coefficient a4

Example 3

(60) FIG. 5 shows an arrangement of an optical scanning system 100C of Example 3. The optical scanning system 100C includes a condenser lens 103C, an aperture 104C, a variable-focus element 105C, an imaging lens 106C and a deflector 107 such as a polygon mirror. Light emitted by a laser diode light source 101, wavelength of which is 780 nm, is converted into a predetermined divergent beam by the condenser lens 103C, made to pass through the variable-focus element 105C and deflected by the deflector 107 such that the beam scans a light receiving surface 108 in the horizontal direction in FIG. 5. Then, the beam passes through the imaging lens 106C. The variable-focus element 105C is controlled such that the beam constantly forms an image on the light receiving surface during the period of the scanning. In FIG. 5, the center of the scanning line on the light receiving surface 108 is represented as C and the both ends are represented as E.

(61) Table 5 shows numerical data of the optical scanning system 100C of Example 3.

(62) The variable-focus element is a virtual one which has a thickness of 0. However, in FIG. 5, the variable-focus element 105C is drawn as one which has a predetermined thickness for the sake of better understanding.

(63) TABLE-US-00005 TABLE 5 Radius of Space or curvature thickness at center Item (mm) (mm) Material Datum of light source 4.719 Entry side surface of 2 17.751 BK7 condenser lens Exit side surface of 1 3.662 condenser lens Aperture plane 3 Principal point on entry 0 side of variable-focus element Principal point on exit 95 side of variable-focus element Deflecting mirror 20 Entry side surface of 4 96.629 COP imaging lens (n = 1.525@780 nm) Exit side surface of 180 36.389 imaging lens Image surface of scanning beam (light receiving surface)

(64) The material of the condenser lens 103C is borosilicate glass, the brand name of which is BK7. For light of wavelength of 780 nm, the value of refractive index is 1.511 and the value of Abbe's number is 64.2. The material of the imaging lens 106C is cyclo-olefin polymer (COP), the brand name of which is ZEONEX E48R. For light of wavelength of 780 nm, the value of refractive index is 1.525 and the value of Abbe's number is 56.0.

(65) The entry side surface of the condenser lens 103C is spherical. The exit side surface of the condenser lens 103C is defined by the following equations.

(66) 0$\begin{array}{cc}z=\frac{{\mathrm{cr}}^2}{1+\sqrt{1-}\left(1+k\right)c^2{r}^2}+\underset{n}{.Math.}{a}_{2n}{r}^{2n}& \left(22\right)\\ r=\sqrt{x^2+y^2}& \left(23\right)\end{array}$

(67) Table 6 shows numerical data of the exit side surface of the condenser lens 103C of Example 3. The focal length of the condenser lens 103C is 6.1 millimeters.

(68) TABLE-US-00006 TABLE 6 Item Data (Coefficient) Radius of curvature 3.662 at center R = 1/c Conic constant k 0.997 Fourth-order aspherical 4.91E04 coefficient a4

(69) The entry side and exit side surfaces of the imaging lens 106C are troidal surfaces which are obtained by rotating the generatrix defined by the following equation with a diameter Rr.

(70) $\begin{array}{cc}z=\frac{{\mathrm{cy}}^2}{1+\sqrt{1-}\left(1+k\right)c^2{y}^2}+\underset{n}{.Math.}{}_{2n}{y}^{2n}& \left(24\right)\end{array}$

(71) Table 7 shows numerical data of the entry side and exit side surfaces of the imaging lens 106C of Example 3. The unit of length in the table is millimeter.

(72) TABLE-US-00007 TABLE 7 Entry side Exit side Item surface surface Radius of rotation Rr 253.425 73.347 Radius of curvature 96.629 36.389 at center of generatrix R = 1/c Conic constant of 0 0.977 generatrix k Fourth-order 3.34E06 6.43E07 aspherical coefficient of generatrix 4 Sixth-order aspherical 0 3.40E09 coefficient of generatrix 6

Comparative Example

(73) FIG. 6 shows an arrangement of an optical scanning system 100X of Comparative Example. The optical scanning system 100X includes a condenser lens 103X, an aperture 104X, a variable-focus element 105X, an imaging lens 106X and a deflector 107 such as a polygon mirror. Light emitted by a laser diode light source 101, wavelength of which is 780 nm, is converted into a predetermined divergent beam by the condenser lens 103X, made to pass through the variable-focus element 105X and the imaging lens 106X and deflected by the deflector 107 such that the beam scans a light receiving surface 108 in the horizontal direction in FIG. 6. The variable-focus element 105X is controlled such that the beam constantly forms an image on the light receiving surface during the period of the scanning. In FIG. 6, the center of the scanning line on the light receiving surface 108 is represented as C and the both ends are represented as E.

(74) Table 8 shows numerical data of the optical scanning system 100X of Comparative Example.

(75) The variable-focus element is a virtual one which has a thickness of 0. However, in FIG. 6, the variable-focus element 105X is drawn as one which has a predetermined thickness for the sake of better understanding.

(76) TABLE-US-00008 TABLE 8 Radius of Space or curvature thickness at center Item (mm) (mm) Material Datum of light source 10 Entry side surface of 3 Flat BK7 condenser lens Exit side surface of 1 6.209 condenser lens Aperture plane 3 Principal point on entry 0 side of variable-focus element Principal point on exit 239 side of variable-focus element Entry side surface of 2 Flat BK7 imaging lens Exit side surface of 25 124.446 imaging lens Deflecting mirror 202.026 Image surface of scanning beam (light receiving surface)

(77) The material of the condenser lens 103X and the imaging lens 106X is borosilicate glass, the brand name of which is BK7. For light of wavelength of 780 nm, the value of refractive index is 1.511 and the value of Abbe's number is 64.2.

(78) The entry side and exit side surfaces of the condenser lens 103X and the entry side and the exit side surfaces of the imaging lens 106X are spherical or flat. The focal length of the condenser lens 103X is 6.1 millimeters.

Comparison Between the Examples and the Comparison Example

(79) Table 9 shows important parameters of Examples 1 to 3 and Comparative Example.

(80) TABLE-US-00009 TABLE 9 Comparative Parameter Example 1 Example 2 Example 3 Example x1 42.2 41.9 131.9 282296.0 x2 135.0 135.2 119.1 240.3 x3 245.2 245.2 191.7 243.2 f_max 257.5 255.0 764.4 4249.6 f_min 257.5 255.0 764.4 4249.6 f 257.5 255.0 764.4 4249.6 f2 102.9 102.9 108.7 243.4 H 108.0 108.0 108.0 108.6 A (corresponding to 321.9 326.1 35.0 3.1 Expression (1)) B (corresponding to 742.6 742.9 481.6 281517.6 Expression (2)) C (corresponding to 61.2 61.0 34.2 45.7 Expression (3)) D (corresponding to 0.071 0.072 0.013 0.001 Expression (5)) 1 206.0 206.3 206.4 197.4 x3+ 231.0 231.0 178.3 229.3 x3 257.6 257.6 204.8 257.2 E (corresponding to 0.010 0.011 0.048 0.067 Expression (6))

(81) By the use of FIG. 2, x.sub.1, x.sub.2, x.sub.3, and f have been defined as below. It is assumed that the reciprocal of the focal length f of the variable-focus element 105 is changed from the minimum value 1/f.sub.MIN to the maximum value 1/f.sub.MAX such that the beam constantly forms an image on the light receiving surface during the period of the scanning. When
1/f={(1/f.sub.MAX)+(1/f.sub.MIN)}/2=1/f.sub.MEAN,
the distance from the virtual image point P1 of the divergent beam to the principal point P2 on the entry side of the variable-focus element 105 is represented as x.sub.1, the distance from the principal point P3 on the exit side of the variable-focus element 105 to the principal point P4 on the entry side of the imaging lens 106 is represented as x.sub.2, and the distance from the principal point P5 on the exit side of the imaging lens 106 to the image point P6 is represented as x.sub.3. Further, f is defined as below.
f.sub.MAX=f
f.sub.MIN=f

(82) Additional description of Example 2 will be given below. The optical scanning system 100B of Example 2 does not include a variable-focus element. In the optical scanning system 100B of Example 2, a position of an image of the beam along the optical path is changed by the movement of the condenser lens 103B in the direction of the optical axis such that the beam constantly forms an image on the light receiving surface during the period of the scanning. In this case, it is assumed that the condenser lens 103B and a virtual variable-focus element 105 are fixed at the center of the section in which the condenser lens 103B moves in the direction of the optical axis, and the minimum value 1/f.sub.MIN and the maximum value 1/f.sub.MAX of the reciprocal of the focal length of the virtual variable-focus element 105 are determined such that the change of the position of the image caused by the movement of the condenser lens 103B in the direction of the optical axis is reproduced. The values of f.sub.MIN (f) and f.sub.MAX (f) in Table 9 are determined as described above.

(83) By the use of FIG. 3, l and H have been defined as below. l represents the distance from the point of reflection on the deflector to the center C of the scanning line along the path of the principal ray of the beam which reaches the center C of the scanning line. H represents the distance from the center C of the scanning line to each end E of the scanning line. In the case of Example 3, however, l represents length of the side along the primary ray of the beam which travels toward the center C of the scanning line of the triangle formed by the scanning line on the light receiving surface, the primary ray of the beam which travels toward the center C of the scanning line and the primary ray of the beam which travels toward one of the ends E of the scanning line.

(84) A to E are defined as below and correspond respectively to Expressions (1) to (3), (5) and (6).

(85) $A=x_2+\frac{x_2^2}{x_1}-x_3$$B=.Math.x_1.Math.-3.2x_3$$C=x_2-0.8x_3$$D=\frac{x_1\left(x_1{x}_2+x_2^2-x_1{x}_3\right)}{f\left(x_1+x_2\right)}$$E=-\frac{{\left(x_{3-}-x_{3+}\right)}^2}{H^2}+\frac{x_1\left(f_2-x_2\right)}{f\left(x_1+x_2-f_2\right)}$

(86) According to Table 9, Examples 1 to 3 satisfy all of Expressions (1) to (3), (5) and (6). On the other hand, Comparative Example does not satisfy any of Expressions (1) to (3), (5) and (6).

(87) FIG. 7 shows a relationship between position and diameter of the spot formed by the beam on the light receiving surface 108. The horizontal axis of FIG. 7 represents distance from the center C to an arbitrary point on the scanning line on the light receiving surface, that is, image height. The unit is millimeter. The direction from the center C toward one of the two ends of the scanning line is defined as a positive direction, and the direction from the center C toward the other of the two ends of the scanning line is defined as a negative direction. The vertical axis of FIG. 7 represents spot diameter. The unit is micrometer.

(88) As to Examples 1 to 3, the values of the variance of spot diameter across the horizontal axis are smaller than 2 micrometers. As to Comparative Example (Conventional Example), the value of the variance of spot diameter across the horizontal axis is greater than 7 micrometers. Thus, the values of the variance of spot diameter in Examples 1 to 3 are remarkably reduced in comparison with the value of the variance of spot diameter in Comparative Example.

(89) In other preferred embodiments, by making the focal length of the imaging lens in the main scanning direction and that in the sub scanning direction different from each other, a linear image can be formed on a reflecting surface of the deflector for optical face tangle error correction, or mirrors of the deflector can be downsized.